Graph y=cot(x-(5pi)/6)

$y = \cot (x - \frac{5 π}{6})$

Find the asymptotes.

For any $y = \cot (x)$ , vertical asymptotes occur at $x = n π$ , where $n$ is an integer. Use the basic period for $y = \cot (x)$ , $(0, π)$ , to find the vertical asymptotes for $y = \cot (x - \frac{5 π}{6})$ . Set the inside of the cotangent function, $b x + c$ , for $y = a \cot (b x + c) + d$ equal to $0$ to find where the vertical asymptote occurs for $y = \cot (x - \frac{5 π}{6})$ .

$x - \frac{5 π}{6} = 0$

Add $\frac{5 π}{6}$ to both sides of the equation.

$x = \frac{5 π}{6}$

Set the inside of the cotangent function $x - \frac{5 π}{6}$ equal to $π$ .

$x - \frac{5 π}{6} = π$

Move all terms not containing $x$ to the right side of the equation.

Add $\frac{5 π}{6}$ to both sides of the equation.

$x = π + \frac{5 π}{6}$

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = π \cdot \frac{6}{6} + \frac{5 π}{6}$

Combine $π$ and $\frac{6}{6}$ .

$x = \frac{π \cdot 6}{6} + \frac{5 π}{6}$

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 + 5 π}{6}$

Simplify the numerator.

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π + 5 π}{6}$

Add $6 π$ and $5 π$ .

$x = \frac{11 π}{6}$

The basic period for $y = \cot (x - \frac{5 π}{6})$ will occur at $(\frac{5 π}{6}, \frac{11 π}{6})$ , where $\frac{5 π}{6}$ and $\frac{11 π}{6}$ are vertical asymptotes.

$(\frac{5 π}{6}, \frac{11 π}{6})$

Find the period $\frac{π}{| b |}$ to find where the vertical asymptotes exist.

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

The vertical asymptotes for $y = \cot (x - \frac{5 π}{6})$ occur at $\frac{5 π}{6}$ , $\frac{11 π}{6}$ , and every $x = \frac{5 π}{6} + π n$ , where $n$ is an integer.

$x = \frac{5 π}{6} + π n$

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{5 π}{6} + π n$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{5 π}{6} + π n$ where $n$ is an integer

Use the form $a \cot (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 1$

$b = 1$

$c = \frac{5 π}{6}$

$d = 0$

Since the graph of the function $c o t$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Find the period of $\cot (x - \frac{5 π}{6})$ .

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

Find the phase shift using the formula $\frac{c}{b}$ .

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{\frac{5 π}{6}}{1}$

Divide $\frac{5 π}{6}$ by $1$ .

Phase Shift: $\frac{5 π}{6}$

List the properties of the trigonometric function.

Amplitude: None

Period: $π$

Phase Shift: $\frac{5 π}{6}$ ( $\frac{5 π}{6}$ to the right)

Vertical Shift: None

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Vertical Asymptotes: $x = \frac{5 π}{6} + π n$ where $n$ is an integer

Amplitude: None

Period: $π$

Phase Shift: $\frac{5 π}{6}$ ( $\frac{5 π}{6}$ to the right)

Vertical Shift: None

Graph y=cot(x-(5pi)/6)

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